Whistler Alley Mathematics

These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.

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Lona
Hecho en EE. UU. o importado
sintético único
Con cada par de zapatos que compra, TOMS & reg; dará un nuevo par de zapatos para un niño necesitado One for One & reg;.
El TOMS & reg; Los clásicos mantendrán eres cálido y tostado cada día frío!
Parte superior viene en una variedad de materiales. declarado en color nombre.)
Características puntada clásica del dedo del pie detallando.
Elástico V-panel para fácil encendido y apagado. faux shearling revestimiento. plantilla proporciona amortiguación ligera. Inserción de arco para soporte agregado Flexible, de una sola pieza suela de caucho mixto. Importado. Peso: 5 oz Producto las medidas fueron tomado usando tamaño 12, ancho D - Medio. Tenga en cuenta que mediciones pueden variar por tamaño
Some of the lessons are accompanied by questions and suggestions for extensions.

In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students’ lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.

I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.

Many of these presentations use the Geometer’s Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.

Paul Kunkel


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Recent Changes

March 16, 2016

Updated the animations on Hanging With Galileo.

October 23, 2015

Updated the animations on Reuleaux Triangle.

October 9, 2015

Updated the animations on the Vernier Scale page of the Geometry of Surveying.

October 8, 2015

TOMS - oliva Slip-On de lienzo clásico para B077XNS5TV 19132 hombre Lienzo verde oliva 0abe4e8

Updated the animations on Linkages.

October 4, 2015

Updated the animations on On the Square.

September 25, 2015

Updated the animations on Tangent Circles.

September 23, 2015

Updated the lead page of Conic Sections and Construction.
The page it replaced had java animations that were not being treated well by modern browsers. New images were generated with the developmental Web Sketchpad, which is producing very satisfying results. This appears to be the future, so expect more of my pages to be similarly updated soon.

February 26, 2015

Added Book VII to the Conics of Apollonius.
This is the final surviving book of the treatise.

October 22, 2014

Added Book VI to the Conics of Apollonius.

June 10, 2014

Added Book V to the Conics of Apollonius.


 Includes GSP-3 files

 Includes GSP-4 files

 Includes GSP-5 files Contents:

The Brachistochrone

We are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time.

This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most.


Buffon’s Needle

An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.


TOMS - oliva Slip-On de lienzo clásico para B077XNS5TV 19132 hombre Lienzo verde oliva 0abe4e8

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

Trotters para mujer Dea Slingback Pump Black Micro / Satin , Pantuflas Wilma de Katy Perry para mujer Negro Trotters para mujer Dea Slingback Pump Negro Negro , Sandalias de vestir Patti de Sam Edelman para mujer Patente negra Strive Footwear Vienna Orthotic Slipper Gris carbón - Nubuck Sandalia de vestir Taimi de Naturalizer para mujer Brillo negro , Trotters para mujer Dea Slingback Pump Black Micro / Satin Zapatillas Nami3 de Easy Spirit para mujer Negro Trotters Lucy Ballet para mujer Estaño Sam Edelman Women s Addison Sandalia de tacón Rosa magenta , Bomba de Angel II para mujeres estilo suave Elegancia Roja Zapatilla Glamour de mujer verde Daniel Gold Kidskin , Trotters Lucy Ballet para mujer Black Suede / Patent Zapatilla de cuña para mujer Jacques Levine # 4640 Hueso El Stella de Stella de El Naturalista Oceano , J.Renee Womens Bilboa Rosa Pastel Multi TOMS Clásicos de mujer Negro sobre lienzo negro , Bomba de Angel II para mujeres estilo suave Elegancia marrón oscuro Sandalia de golf Sandbaggers Carrie para mujer rojo Bomba Parigi de LifeStride para mujer Gris pardo , Egos Copenhagen House Slip on Slippers Rusty Red JOE N JOYCE Paris Sandalias de suela suave SynSoft Florigreen TOMS - Slip-On de lienzo clásico para hombre Negro sobre negro Cuello de piel con cuentas y conejo de piel de conejo Laurentian, pantuflas de ante negro CLARKS Keesha Rosa Dress Pump para mujer Dusty Pink Leather Zapatillas Shirley con forro de piel de oveja Overland Shirley para mujer castaña Zapatillas terapéuticas para mujer Cuddle Dr.comfort - Rosa Rosado , Bomba de vestido Daisie de mujer Steve Madden Ante azul Minnetonka Womens Chrissy Slipper Bootie Canela
trigonometry, and probability. Fair warning: The discussion revolves about a figure that appears only when the sketch is animated. Without Geometer’s Sketchpad, this one will make little sense at all.


Conic Sections and Construction

My emphasis is on geometric properties of the conic sections, particulary understanding that has been handed down to us in ancient texts. There also are some challenging constructions.


The Geometry of Surveying

This used to be my job, and I enjoyed studying all of the geometrical concepts that were part of the science of land surveying. In this article I discuss the instruments of the profession, their operating principles, their error, and their calibration.


Hanging with Galileo

An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.


Inversion Geometry Mizuno Women s Wave Prophecy 5-w Gunmetal / rosa , Sandalia original Teva Women s Original Campo Negro / Blanco , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Black / Black / Tech Rust , Sandalias con cordones Teva Original Original Rope Women de Teva Original Morado oscuro Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Hyper Cobalt / White , Zapatillas de running Nike Air Zoom Elite 8 para mujer Racer Blue / Hypr Orange / Sl / Blck La Sportiva Akyra GTX Running Shoe - Mujer Pizarra / Azufre Calzado de trail running In-X-Talon 212 (S) para hombres Inov-8 Negro / Rojo Zapatillas de running Saucony para mujer Omni 16 Gris azul , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar 212blackwhite , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Mushroom Nubuck , Earth Origins Hadley Hibiscus Red , Zapatillas de running Cavu para mujer HOKA ONE ONE Vintage Indigo / Hibiscus , Zapatillas de running NIKE Air Max 90 EZ para mujer Blanco Blanco , Orthofeet 851 Comfort Diabetic Therapeutic Profth Shoes para mujer , Calzado deportivo de mujer Fila de Statique Dark Shade / Black / Purple Cactus Flower Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco / rojo Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Moca , Trotters Mujer Simmy Mary Jane Flat Azul oscuro , Sandalia para mujer Qtee Slide de Anne Klein Negro sintético Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Platino , Noble Outfitters Mens todo alrededor del dedo del pie cuadrado resistente Negro Bota occidental con flecos y tachuelas Roper para mujer Black / Crystal Stud , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Llovizna yute metálico gris Pantuflas Tashia de Katy Perry para mujer Uva Zapato clásico Haflinger para mujer Negro Bernie Mev Women s Gem Wedge Pump Jeans Combo Zapatillas Nike Women s Flex Trainer 5 Negro / antracita / blanco , ara Mujer Prim Oxford Flat Pebble Suede / Ruvido Kmeioo Slip On Mocasines, mujeres Bordado mula Zapatos con felpa Terciopelo de piel de cordero Sin respaldo Toe Mule Slides Tacón negro sólido

If you have never studied inversion geometry, you can start here. The elementary concepts are explained. This page is a quick reference for the other lessons that employ inversion geometry.


Linkages

Draw a line segment. Did you use a straightedge? How do you know it was straight?

Java simulations are used to investigate a nineteenth century mechanics problem.


On the Square

The challenge: Given four arbitrary, unique points, A, B, C, and D, construct one line through each point such that the four lines form a square.


Orthocenter What is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers’ workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer’s Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella.
The Planimeter

A planimeter is a drafting instrument used to measure area. Its mechanism is quite simple, but understanding why it works is more elusive.


Platonic Solids

Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.


Reuleaux Triangle

It is not a circle, but it has constant width. Study the properties of this figure with interactive animated sketches.


Rungs and Vineyards

How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.


The Sliding Triangle

An interesting accidental discovery – Investigate ellipses by tracing triangle transformations.


Table Patterns

Explore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic.

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Tangent Circles

Three circles are given. How many circles can be constructed tangent to all three? How are they constructed? This lesson includes ten animations covering all of the special cases. Some of them are very complex. A little knowledge of circle inversions would help.


Volume of a Torus Do you already know the volume of a torus? Then humor me. I just figured TOMS - oliva Slip-On de lienzo clásico para B077XNS5TV 19132 hombre Lienzo verde oliva 0abe4e8 it out, and I got a chance to try out this new graphics software.


Resources


The Geometer’s Sketchpad Workshop

It is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad.

Zapatillas
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

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and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed.

Back to Whistler Alley


Last update: March 16, 2016 ... Paul Kunkel whistling@whistleralley.com
For email to reach me, the word geometry must appear in the body of the message.